IF THERE ARE 'N' NUMBER OF PEOPLE WHAT IS THE PROBABILITY THAT TWO PEOPLE WILL HAVE BIRTHDAY ON SAME DAY

This problem has been solved with an assumption that all years have 365 days, no leap years.

Instead of calculating the probability of at least 2 people in the group of N people having their birthday on the same day, we can calculate the probability that none of them have a birthday on the same day and then subtract that from 1.

Let's start with 2 people. The probability that they don't have a common birthday is (365/365)*(364/365). This gives the probability that they have the birthday on the same day as 1 - (365/365)*(364/365) = 1 - (365*364)/(365)^2

Next take 3 people. The probability that at least 2 share a common birthday is 1-(365*364*363)/(365)^3

Continuing, for N people the probability that at least have the birthday on the same day is:

1 - (365*363*363*...*(365 - n +1))/365^n

=> 1 - 365!/ [(365 - n)!*365^n]

The required probability that two people in a group of N have the birthday on the same day is 1 - 365!/ [(365 - n)!*365^n]